We estimated the equilibrium OMAD [operating (profit) margin *after* depreciation] of certain U.S. retailers using an autoregressive (AR(1)) model built-in RoyaltyStat®. We use the Gauss run-time engine, so the regression estimates are reliable.

**Fixed-Point Equilibrium**

We used an AR(1) model of company OMAD (which we call Y):

(1) Y(*t* + 1) = α + β Y(*t*) + U(*t*),

where Y(*t*) is stable when the absolute value of the slope coefficient is lower than one (|β| < 1). Among others, Walter Enders, *Applied Econometric Time Series*, John Wiley & Sons, 1995, is a useful reference. We overfitted the model to AR(2), testing for serial correlation among the residuals, and report the selected lag on the table below.

The error term U(*t*) is assumed to have zero-mean and constant variance, σ^{2}, and the year index includes all the data available for the lag-period *t* = 1, 2, 3, …, T.

We have criticized the OECD TNMM convention that T = 3 years, including the current audit year and two prior years. We get more reliable results using as many years of OMAD data as available, except that we should exclude outliers because they distort the point estimates and their standard errors.

If μ is the mean (average) OMAD, and the operator E is *expected* *value*, we can obtain a fixed-point equilibrium for the selected profit indicator:

(2) E(Y(*t* + 1)) = α + β E(Y(*t*)), where E(U(*t*)) = 0.

(3) μ = α + β μ from which we get the fixed-point equilibrium:

(4) μ = α / (1 − β), which is positive if the intercept of (1) is positive.

If the intercept α = 0, then μ = 0 and Y(*t*) fluctuates around zero. The intercept represents the routine OMAD expected to be earned by every company in the industry; the slope coefficient is the non-routine OMAD multiplier.

In economics, the non-routine return is attributed to economic concentration (measured by significant market-share) in competition policy, intangibles in transfer pricing, and significant risk in the financial literature. For application of AR(1) to historical company data in various countries, see Dennis Mueller (ed.), *The Dynamics of Company Profits*, Cambridge University Press, 1990, including empirical chapters about listed companies in Canada, France, Germany, Japan, US and UK.

We introduced AR(*p*) models (for *p* = 1, 2) in transfer pricing on Nov. 5, 1991 on an internal memorandum to Charles Triplett, then IRS Deputy Associate Chief Counsel (International), during my participation in the drafting of the U.S. transfer pricing regulations. We can utilize this AR(1) model given the reliability conditions required under 26 CFR §1.482-1(e)(2)(iii)(B), which provides:

“The interquartile range ordinarily provides an acceptable measure of this range; however *a different statistical method may be applied if it provides a more reliable measure *[of the arm's length range].”

Suppose that a U.S. controlled retailer (tested party) under audit has 13 comparables selected from a list of 21 retailers. The table below summarizes our empirical results of the equilibrium OMAD computed using equation (4). GVKEY is Standard & Poor's Compustat “Global Vantage Key” that is unique for every company. Count (sample size) is the number of annual OMAD terminating in 2016, going back various years as the data are available per company:

Company | GVKEY | Lag | Count |
Equilibrium OMAD (%) |
Std. Error OMAD (%) |
Std. Error Residuals (%) |

Belk Inc. | 117981 | 2 | 17 | 6.9 | 1.1 | 1.1 |

Best Buy | 2184 | 2 | 33 | 3.7 | 1.0 | 1.0 |

Bon Ton | 24483 | 1 | 26 | 3.2 | 1.4 | 1.4 |

Conn's Inc. | 156614 | 1 | 14 | 6.8 | 2.9 | 2.8 |

Costco | 29028 | 1 | 25 | 2.9 | 0.2 | 0.2 |

Home Depot | 5680 | 1 | 37 | 12.3 | 1.3 | 1.2 |

J.C. Penneys | 8446 | 2 | 34 | 5.9 | 1.4 | 1.3 |

Kohl's Corp. | 25283 | 1 | 26 | 9.6 | 0.9 | 0.9 |

Lowe's | 6829 | 2 | 37 | 7.6 | 0.8 | 0.7 |

Macy's | 4611 | 1 | 38 | 6.9 | 1.1 | 1.1 |

PriceSmart Inc. | 65343 | 1 | 13 | 5.2 | 0.4 | 0.4 |

Target Corp. | 3813 | 1 | 38 | 6.7 | 0.7 | 0.7 |

Wal-Mart | 11259 | 2 | 37 | 5.2 | 0.3 | 0.3 |

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The standard error of the equilibrium OMAD is small compared to the equilibrium OMAD, suggesting that this AR(1) model has a major advantage in producing low coefficients of variation. E.g., counting 33 years of data, Best Buy has an equilibrium OMAD = 3.7% ± 1.0%. Low coefficients of variation imply reliable measures of the variable (OMAD) of interest, because the reliability of the estimated mean or expected value depends on the standard deviation. The coefficient of variation is the ratio of the standard deviation to the mean and measures the relative reliability of the sample mean.

**Variance of Fixed-Point Equilibrium**

Like the derivation of the equilibrium point, we derive the standard error of the equilibrium position by letting V denote the variance operator, and then assuming a constant or stable variance over time:

(5) V(Y(*t* + 1)) = V(Y(*t* )) = Σ.

Applying this time-invariant measure of data spread (Σ) to equation (1), we obtain (ignoring covariance for the purpose of exposition but not of calculation):

(6) Σ = β^{2} Σ + σ^{2}, which yields the equilibrium variance:

(7) Σ = σ^{2} / (1 − β^{2}) for β < 1.

The standard error of the equilibrium position (4) is the square root of (8) after adding the covariance term. We know from first-order difference equations like AR(1) that we get stable results when β < 1. Else, if β ≥ 1, the variance grows exponentially with time, and there is no finite limit for long horizons, *i.e*., we get no fixed-point equilibrium value.

We conclude that an AR(1) or AR(2) model can produce reliable estimates of OMAD, which we can compare to the results of the naive model applied by rote to three years of data as provided on the OECD *Transfer Pricing* *Guidelines *(1995, 2017) and determine what's the best fit of the comparable data under the facts and circumstances. Naivete is not a reliable saddle to ride in a controversial arena such as arm's length income tax determination.

**Ednaldo Silva**(Ph.D.) is founder and managing director at RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at: esilva@royaltystat.com

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